As my Right Hon Friend the Prime Minister set out to the House (https://hansard.parliament.uk/commons/2020-02-03/debates/20020316000004/DepartmentForExitingTheEuropeanUnionClosure) the Department for Exiting the European Union (DExEU) closed on 31 January. Those of its functions which are still required have been transferred to relevant government departments. The administration of the closure of DExEU is being carried out by the Cabinet Office.
The Government department responsible for answering written questions will depend on the subject of the question. Questions can be transferred between departments to ensure that they are answered by the relevant department. The Government department responsible for answering oral questions will depend on the subject of the question.
Once the Exiting the European Union Select Committee members are nominated and it is fully constituted, the Committee may wish, in the first instance, to take up any questions it has with the Cabinet Office.